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2.2: The Inverse of a Matrix
https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/02%3A_Matrices/2.02%3A_The_Inverse_of_a_Matrix
This page explores matrix operations, focusing on the identity matrix and matrix inverses, including their existence, uniqueness, and the method for finding inverses through augmented matrices and row...This page explores matrix operations, focusing on the identity matrix and matrix inverses, including their existence, uniqueness, and the method for finding inverses through augmented matrices and row operations. It provides examples illustrating both the derivation of inverses and scenarios where matrices lack inverses.
3.6: What Are the Odds?
https://math.libretexts.org/Courses/Florida_SouthWestern_State_College/MGF_1131%3A_Mathematics_in_Context__(FSW)/03%3A_Counting_and_Probability/3.06%3A_What_Are_the_Odds
This section explains how to compute and convert odds and probabilities, particularly in contexts like lotteries and weather predictions. It defines odds in favor and against, illustrating how to expr...This section explains how to compute and convert odds and probabilities, particularly in contexts like lotteries and weather predictions. It defines odds in favor and against, illustrating how to express them as ratios. The formula for calculating the probability from given odds is provided, along with examples and exercises to enhance understanding. The importance of differentiating between odds and probabilities in communication is underscored throughout.
8.3: Mental Arithmetic—Using the Distributive Property
https://math.libretexts.org/Bookshelves/PreAlgebra/Fundamentals_of_Mathematics_(Burzynski_and_Ellis)/08%3A_Techniques_of_Estimation/8.03%3A_Mental_ArithmeticUsing_the_Distributive_Property
\(\begin{array} {rclc} {3(2 + 5)} & = & {\underbrace{2 + 5 + 2 + 5 + 2 + 5}_{\text{2 + 5 appears 3 times}}} & {} \\ {} & = & {2 + 2 + 2 + 5 + 5 + 5} & {\text{(by the commutative property of addition)}... $\begin{array} {rclc} {3(2 + 5)} & = & {\underbrace{2 + 5 + 2 + 5 + 2 + 5}_{\text{2 + 5 appears 3 times}}} & {} \\ {} & = & {2 + 2 + 2 + 5 + 5 + 5} & {\text{(by the commutative property of addition)}} \\ {} & = & {3 \cdot 2 + 3 \cdot 5} & {\text{(since multiplication describes repeated addition)}} \\ {} & = & {6 + 15} & {} \\ {} & = & {21} & {} \end{array}$
1.4: Addition and Subtraction in Base Systems
https://math.libretexts.org/Courses/Florida_SouthWestern_State_College/MGF_1131%3A_Mathematics_in_Context__(FSW)/01%3A__Number_Representation_in_Different_Bases_and_Cryptography/1.04%3A__Addition_and_Subtraction_in_Base_Systems
This section focuses on arithmetic in different numeral systems, specifically bases 2 through 9 and 12. It explains how computers use base 2 (binary) for calculations and how conventional base 10 arit...This section focuses on arithmetic in different numeral systems, specifically bases 2 through 9 and 12. It explains how computers use base 2 (binary) for calculations and how conventional base 10 arithmetic changes with non-decimal bases. It details the construction of addition tables for these bases and illustrates addition and subtraction through examples. Key examples include calculations in bases 6, 7, and 12, and it also highlights common errors.
4: Determinants
https://math.libretexts.org/Bookshelves/Linear_Algebra/Interactive_Linear_Algebra_(Margalit_and_Rabinoff)/04%3A_Determinants
This page discusses matrix equations, focusing on solving $Ax=b$ , determining eigenvalues and eigenvectors, and finding approximate solutions. The current chapter emphasizes determinants, covering t...This page discusses matrix equations, focusing on solving $Ax=b$ , determining eigenvalues and eigenvectors, and finding approximate solutions. The current chapter emphasizes determinants, covering their definition, properties, and computation methods. It includes cofactor expansions as a recursive calculation method and explores the geometric interpretation of determinants in relation to volumes in multivariable calculus.
8.4: Estimation by Rounding Fractions
https://math.libretexts.org/Bookshelves/PreAlgebra/Fundamentals_of_Mathematics_(Burzynski_and_Ellis)/08%3A_Techniques_of_Estimation/8.04%3A_Estimation_by_Rounding_Fractions
Estimation by rounding fractions is a useful technique for estimating the result of a computation involving fractions. Fractions are commonly rounded to 1/4, 1/2, 3/4, 0, and 1. Remember that round...Estimation by rounding fractions is a useful technique for estimating the result of a computation involving fractions. Fractions are commonly rounded to 1/4, 1/2, 3/4, 0, and 1. Remember that rounding may cause estimates to vary.
6.4: Orthogonal Sets
https://math.libretexts.org/Bookshelves/Linear_Algebra/Interactive_Linear_Algebra_(Margalit_and_Rabinoff)/06%3A_Orthogonality/6.04%3A_The_Method_of_Least_Squares
This page covers orthogonal projections in vector spaces, detailing the advantages of orthogonal sets and defining the simpler Projection Formula applicable with orthogonal bases. It includes examples...This page covers orthogonal projections in vector spaces, detailing the advantages of orthogonal sets and defining the simpler Projection Formula applicable with orthogonal bases. It includes examples of projecting vectors onto subspaces, emphasizes the importance of orthogonal bases, and introduces the Gram-Schmidt process for generating orthogonal bases from sets of vectors.
8.1: Estimation by Rounding
https://math.libretexts.org/Bookshelves/PreAlgebra/Fundamentals_of_Mathematics_(Burzynski_and_Ellis)/08%3A_Techniques_of_Estimation/8.01%3A_Estimation_by_Rounding
The rounding that is done in estimation does not always follow the rules of rounding discussed in [link] (Rounding Whole Numbers). The rounding technique estimates the result of a computation by round...The rounding that is done in estimation does not always follow the rules of rounding discussed in [link] (Rounding Whole Numbers). The rounding technique estimates the result of a computation by rounding the numbers involved in the computation to one or two nonzero digits. The product can be estimated by $70 \cdot 50 = 3,500$ . (Recall that to multiply numbers ending in zeros, we multiply the nonzero digits and affix to this product the total number of ending zeros in the factors.
3.2: Grouping Symbols and the Order of Operations
https://math.libretexts.org/Bookshelves/PreAlgebra/Fundamentals_of_Mathematics_(Burzynski_and_Ellis)/03%3A_Exponents_Roots_and_Factorization_of_Whole_Numbers/3.02%3A_Grouping_Symbols_and_the_Order_of_Operations
\(\begin{array} {ll} {6 \cdot (3^2 + 2^2) + 4^2} & {\text{ Evaluate the exponential forms in the parentheses: } 3^2 = 9 \text{ and } 2^2 = 4} \\ {6 \cdot (9 + 4) + 4^2} & {\text{ Add the 9 and 4 in th... $\begin{array} {ll} {6 \cdot (3^2 + 2^2) + 4^2} & {\text{ Evaluate the exponential forms in the parentheses: } 3^2 = 9 \text{ and } 2^2 = 4} \\ {6 \cdot (9 + 4) + 4^2} & {\text{ Add the 9 and 4 in the parentheses: 9 + 4 = 13}} \\ {6 \cdot (13) + 4^2} & {\text{ Evaluate the exponential form: } 4^2 = 16} \\ {6 \cdot (13) + 16} & {\text{ Multiply 6 and 13: } 6 \cdot 13 = 78} \\ {78 + 16} & {\text{ Add 78 and 16: 78 + 16 = 94}} \\ {94} & {} \end{array}$

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